3.1853 \(\int (A+B x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=452 \[ -\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{11 e^7 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{9 e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{7 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5 (B d-A e)}{5 e^7 (a+b x)}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (-5 a B e-A b e+6 b B d)}{15 e^7 (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{13 e^7 (a+b x)}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^7 (a+b x)} \]

[Out]

(2*(b*d - a*e)^5*(B*d - A*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e
^7*(a + b*x)) - (2*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)^(7/2)*Sqr
t[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) + (10*b*(b*d - a*e)^3*(3*b*B*d - 2
*A*b*e - a*B*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x))
- (20*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) + (10*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2
*a*B*e)*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)) - (2*
b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(15*e^7*(a + b*x)) + (2*b^5*B*(d + e*x)^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(1
7*e^7*(a + b*x))

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Rubi [A]  time = 0.705311, antiderivative size = 452, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ -\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{11 e^7 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{9 e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{7 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5 (B d-A e)}{5 e^7 (a+b x)}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (-5 a B e-A b e+6 b B d)}{15 e^7 (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{13 e^7 (a+b x)}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(2*(b*d - a*e)^5*(B*d - A*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e
^7*(a + b*x)) - (2*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)^(7/2)*Sqr
t[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) + (10*b*(b*d - a*e)^3*(3*b*B*d - 2
*A*b*e - a*B*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x))
- (20*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) + (10*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2
*a*B*e)*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)) - (2*
b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(15*e^7*(a + b*x)) + (2*b^5*B*(d + e*x)^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(1
7*e^7*(a + b*x))

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Rubi in Sympy [A]  time = 75.1397, size = 452, normalized size = 1. \[ \frac{B \left (2 a + 2 b x\right ) \left (d + e x\right )^{\frac{5}{2}} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{17 b e} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}} \left (17 A b e - 5 B a e - 12 B b d\right )}{255 b e^{2}} + \frac{4 \left (5 a + 5 b x\right ) \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (17 A b e - 5 B a e - 12 B b d\right )}{3315 b e^{3}} + \frac{32 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (17 A b e - 5 B a e - 12 B b d\right )}{7293 b e^{4}} + \frac{64 \left (3 a + 3 b x\right ) \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (17 A b e - 5 B a e - 12 B b d\right )}{65637 b e^{5}} + \frac{256 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (17 A b e - 5 B a e - 12 B b d\right )}{153153 b e^{6}} + \frac{512 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (17 A b e - 5 B a e - 12 B b d\right )}{765765 b e^{7} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

B*(2*a + 2*b*x)*(d + e*x)**(5/2)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(17*b*e) +
2*(d + e*x)**(5/2)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)*(17*A*b*e - 5*B*a*e - 12*
B*b*d)/(255*b*e**2) + 4*(5*a + 5*b*x)*(d + e*x)**(5/2)*(a*e - b*d)*(a**2 + 2*a*b
*x + b**2*x**2)**(3/2)*(17*A*b*e - 5*B*a*e - 12*B*b*d)/(3315*b*e**3) + 32*(d + e
*x)**(5/2)*(a*e - b*d)**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)*(17*A*b*e - 5*B*a*
e - 12*B*b*d)/(7293*b*e**4) + 64*(3*a + 3*b*x)*(d + e*x)**(5/2)*(a*e - b*d)**3*s
qrt(a**2 + 2*a*b*x + b**2*x**2)*(17*A*b*e - 5*B*a*e - 12*B*b*d)/(65637*b*e**5) +
256*(d + e*x)**(5/2)*(a*e - b*d)**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(17*A*b*e
- 5*B*a*e - 12*B*b*d)/(153153*b*e**6) + 512*(d + e*x)**(5/2)*(a*e - b*d)**5*sqrt
(a**2 + 2*a*b*x + b**2*x**2)*(17*A*b*e - 5*B*a*e - 12*B*b*d)/(765765*b*e**7*(a +
b*x))

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Mathematica [A]  time = 0.943818, size = 491, normalized size = 1.09 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{5/2} \left (21879 a^5 e^5 (7 A e-2 B d+5 B e x)+12155 a^4 b e^4 \left (9 A e (5 e x-2 d)+B \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )-2210 a^3 b^2 e^3 \left (3 B \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )-11 A e \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )+510 a^2 b^3 e^2 \left (13 A e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+B \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )-85 a b^4 e \left (B \left (256 d^5-640 d^4 e x+1120 d^3 e^2 x^2-1680 d^2 e^3 x^3+2310 d e^4 x^4-3003 e^5 x^5\right )-3 A e \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )+b^5 \left (17 A e \left (-256 d^5+640 d^4 e x-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3-2310 d e^4 x^4+3003 e^5 x^5\right )+3 B \left (1024 d^6-2560 d^5 e x+4480 d^4 e^2 x^2-6720 d^3 e^3 x^3+9240 d^2 e^4 x^4-12012 d e^5 x^5+15015 e^6 x^6\right )\right )\right )}{765765 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(5/2)*(21879*a^5*e^5*(-2*B*d + 7*A*e + 5*B*e*x) +
12155*a^4*b*e^4*(9*A*e*(-2*d + 5*e*x) + B*(8*d^2 - 20*d*e*x + 35*e^2*x^2)) - 22
10*a^3*b^2*e^3*(-11*A*e*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 3*B*(16*d^3 - 40*d^2*e
*x + 70*d*e^2*x^2 - 105*e^3*x^3)) + 510*a^2*b^3*e^2*(13*A*e*(-16*d^3 + 40*d^2*e*
x - 70*d*e^2*x^2 + 105*e^3*x^3) + B*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x^2 - 8
40*d*e^3*x^3 + 1155*e^4*x^4)) - 85*a*b^4*e*(-3*A*e*(128*d^4 - 320*d^3*e*x + 560*
d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4) + B*(256*d^5 - 640*d^4*e*x + 1120*d^
3*e^2*x^2 - 1680*d^2*e^3*x^3 + 2310*d*e^4*x^4 - 3003*e^5*x^5)) + b^5*(17*A*e*(-2
56*d^5 + 640*d^4*e*x - 1120*d^3*e^2*x^2 + 1680*d^2*e^3*x^3 - 2310*d*e^4*x^4 + 30
03*e^5*x^5) + 3*B*(1024*d^6 - 2560*d^5*e*x + 4480*d^4*e^2*x^2 - 6720*d^3*e^3*x^3
+ 9240*d^2*e^4*x^4 - 12012*d*e^5*x^5 + 15015*e^6*x^6))))/(765765*e^7*(a + b*x))

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Maple [A]  time = 0.014, size = 689, normalized size = 1.5 \[{\frac{90090\,B{x}^{6}{b}^{5}{e}^{6}+102102\,A{x}^{5}{b}^{5}{e}^{6}+510510\,B{x}^{5}a{b}^{4}{e}^{6}-72072\,B{x}^{5}{b}^{5}d{e}^{5}+589050\,A{x}^{4}a{b}^{4}{e}^{6}-78540\,A{x}^{4}{b}^{5}d{e}^{5}+1178100\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}-392700\,B{x}^{4}a{b}^{4}d{e}^{5}+55440\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+1392300\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}-428400\,A{x}^{3}a{b}^{4}d{e}^{5}+57120\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+1392300\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}-856800\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+285600\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}-40320\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+1701700\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}-928200\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+285600\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}-38080\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+850850\,B{x}^{2}{a}^{4}b{e}^{6}-928200\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+571200\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}-190400\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+26880\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+1093950\,Ax{a}^{4}b{e}^{6}-972400\,Ax{a}^{3}{b}^{2}d{e}^{5}+530400\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}-163200\,Axa{b}^{4}{d}^{3}{e}^{3}+21760\,Ax{b}^{5}{d}^{4}{e}^{2}+218790\,Bx{a}^{5}{e}^{6}-486200\,Bx{a}^{4}bd{e}^{5}+530400\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}-326400\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+108800\,Bxa{b}^{4}{d}^{4}{e}^{2}-15360\,Bx{b}^{5}{d}^{5}e+306306\,A{a}^{5}{e}^{6}-437580\,Ad{e}^{5}{a}^{4}b+388960\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}-212160\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+65280\,Aa{b}^{4}{d}^{4}{e}^{2}-8704\,A{b}^{5}{d}^{5}e-87516\,Bd{e}^{5}{a}^{5}+194480\,B{a}^{4}b{d}^{2}{e}^{4}-212160\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+130560\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}-43520\,Ba{b}^{4}{d}^{5}e+6144\,B{b}^{5}{d}^{6}}{765765\,{e}^{7} \left ( bx+a \right ) ^{5}} \left ( ex+d \right ) ^{{\frac{5}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

2/765765*(e*x+d)^(5/2)*(45045*B*b^5*e^6*x^6+51051*A*b^5*e^6*x^5+255255*B*a*b^4*e
^6*x^5-36036*B*b^5*d*e^5*x^5+294525*A*a*b^4*e^6*x^4-39270*A*b^5*d*e^5*x^4+589050
*B*a^2*b^3*e^6*x^4-196350*B*a*b^4*d*e^5*x^4+27720*B*b^5*d^2*e^4*x^4+696150*A*a^2
*b^3*e^6*x^3-214200*A*a*b^4*d*e^5*x^3+28560*A*b^5*d^2*e^4*x^3+696150*B*a^3*b^2*e
^6*x^3-428400*B*a^2*b^3*d*e^5*x^3+142800*B*a*b^4*d^2*e^4*x^3-20160*B*b^5*d^3*e^3
*x^3+850850*A*a^3*b^2*e^6*x^2-464100*A*a^2*b^3*d*e^5*x^2+142800*A*a*b^4*d^2*e^4*
x^2-19040*A*b^5*d^3*e^3*x^2+425425*B*a^4*b*e^6*x^2-464100*B*a^3*b^2*d*e^5*x^2+28
5600*B*a^2*b^3*d^2*e^4*x^2-95200*B*a*b^4*d^3*e^3*x^2+13440*B*b^5*d^4*e^2*x^2+546
975*A*a^4*b*e^6*x-486200*A*a^3*b^2*d*e^5*x+265200*A*a^2*b^3*d^2*e^4*x-81600*A*a*
b^4*d^3*e^3*x+10880*A*b^5*d^4*e^2*x+109395*B*a^5*e^6*x-243100*B*a^4*b*d*e^5*x+26
5200*B*a^3*b^2*d^2*e^4*x-163200*B*a^2*b^3*d^3*e^3*x+54400*B*a*b^4*d^4*e^2*x-7680
*B*b^5*d^5*e*x+153153*A*a^5*e^6-218790*A*a^4*b*d*e^5+194480*A*a^3*b^2*d^2*e^4-10
6080*A*a^2*b^3*d^3*e^3+32640*A*a*b^4*d^4*e^2-4352*A*b^5*d^5*e-43758*B*a^5*d*e^5+
97240*B*a^4*b*d^2*e^4-106080*B*a^3*b^2*d^3*e^3+65280*B*a^2*b^3*d^4*e^2-21760*B*a
*b^4*d^5*e+3072*B*b^5*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5

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Maxima [A]  time = 0.736415, size = 1243, normalized size = 2.75 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)*(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

2/45045*(3003*b^5*e^7*x^7 - 256*b^5*d^7 + 1920*a*b^4*d^6*e - 6240*a^2*b^3*d^5*e^
2 + 11440*a^3*b^2*d^4*e^3 - 12870*a^4*b*d^3*e^4 + 9009*a^5*d^2*e^5 + 231*(16*b^5
*d*e^6 + 75*a*b^4*e^7)*x^6 + 63*(b^5*d^2*e^5 + 350*a*b^4*d*e^6 + 650*a^2*b^3*e^7
)*x^5 - 35*(2*b^5*d^3*e^4 - 15*a*b^4*d^2*e^5 - 1560*a^2*b^3*d*e^6 - 1430*a^3*b^2
*e^7)*x^4 + 5*(16*b^5*d^4*e^3 - 120*a*b^4*d^3*e^4 + 390*a^2*b^3*d^2*e^5 + 14300*
a^3*b^2*d*e^6 + 6435*a^4*b*e^7)*x^3 - 3*(32*b^5*d^5*e^2 - 240*a*b^4*d^4*e^3 + 78
0*a^2*b^3*d^3*e^4 - 1430*a^3*b^2*d^2*e^5 - 17160*a^4*b*d*e^6 - 3003*a^5*e^7)*x^2
+ (128*b^5*d^6*e - 960*a*b^4*d^5*e^2 + 3120*a^2*b^3*d^4*e^3 - 5720*a^3*b^2*d^3*
e^4 + 6435*a^4*b*d^2*e^5 + 18018*a^5*d*e^6)*x)*sqrt(e*x + d)*A/e^6 + 2/765765*(4
5045*b^5*e^8*x^8 + 3072*b^5*d^8 - 21760*a*b^4*d^7*e + 65280*a^2*b^3*d^6*e^2 - 10
6080*a^3*b^2*d^5*e^3 + 97240*a^4*b*d^4*e^4 - 43758*a^5*d^3*e^5 + 3003*(18*b^5*d*
e^7 + 85*a*b^4*e^8)*x^7 + 231*(3*b^5*d^2*e^6 + 1360*a*b^4*d*e^7 + 2550*a^2*b^3*e
^8)*x^6 - 63*(12*b^5*d^3*e^5 - 85*a*b^4*d^2*e^6 - 11900*a^2*b^3*d*e^7 - 11050*a^
3*b^2*e^8)*x^5 + 35*(24*b^5*d^4*e^4 - 170*a*b^4*d^3*e^5 + 510*a^2*b^3*d^2*e^6 +
26520*a^3*b^2*d*e^7 + 12155*a^4*b*e^8)*x^4 - 5*(192*b^5*d^5*e^3 - 1360*a*b^4*d^4
*e^4 + 4080*a^2*b^3*d^3*e^5 - 6630*a^3*b^2*d^2*e^6 - 121550*a^4*b*d*e^7 - 21879*
a^5*e^8)*x^3 + 3*(384*b^5*d^6*e^2 - 2720*a*b^4*d^5*e^3 + 8160*a^2*b^3*d^4*e^4 -
13260*a^3*b^2*d^3*e^5 + 12155*a^4*b*d^2*e^6 + 58344*a^5*d*e^7)*x^2 - (1536*b^5*d
^7*e - 10880*a*b^4*d^6*e^2 + 32640*a^2*b^3*d^5*e^3 - 53040*a^3*b^2*d^4*e^4 + 486
20*a^4*b*d^3*e^5 - 21879*a^5*d^2*e^6)*x)*sqrt(e*x + d)*B/e^7

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Fricas [A]  time = 0.294985, size = 1145, normalized size = 2.53 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)*(e*x + d)^(3/2),x, algorithm="fricas")

[Out]

2/765765*(45045*B*b^5*e^8*x^8 + 3072*B*b^5*d^8 + 153153*A*a^5*d^2*e^6 - 4352*(5*
B*a*b^4 + A*b^5)*d^7*e + 32640*(2*B*a^2*b^3 + A*a*b^4)*d^6*e^2 - 106080*(B*a^3*b
^2 + A*a^2*b^3)*d^5*e^3 + 97240*(B*a^4*b + 2*A*a^3*b^2)*d^4*e^4 - 43758*(B*a^5 +
5*A*a^4*b)*d^3*e^5 + 3003*(18*B*b^5*d*e^7 + 17*(5*B*a*b^4 + A*b^5)*e^8)*x^7 + 2
31*(3*B*b^5*d^2*e^6 + 272*(5*B*a*b^4 + A*b^5)*d*e^7 + 1275*(2*B*a^2*b^3 + A*a*b^
4)*e^8)*x^6 - 63*(12*B*b^5*d^3*e^5 - 17*(5*B*a*b^4 + A*b^5)*d^2*e^6 - 5950*(2*B*
a^2*b^3 + A*a*b^4)*d*e^7 - 11050*(B*a^3*b^2 + A*a^2*b^3)*e^8)*x^5 + 35*(24*B*b^5
*d^4*e^4 - 34*(5*B*a*b^4 + A*b^5)*d^3*e^5 + 255*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^6
+ 26520*(B*a^3*b^2 + A*a^2*b^3)*d*e^7 + 12155*(B*a^4*b + 2*A*a^3*b^2)*e^8)*x^4 -
5*(192*B*b^5*d^5*e^3 - 272*(5*B*a*b^4 + A*b^5)*d^4*e^4 + 2040*(2*B*a^2*b^3 + A*
a*b^4)*d^3*e^5 - 6630*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^6 - 121550*(B*a^4*b + 2*A*a^
3*b^2)*d*e^7 - 21879*(B*a^5 + 5*A*a^4*b)*e^8)*x^3 + 3*(384*B*b^5*d^6*e^2 + 51051
*A*a^5*e^8 - 544*(5*B*a*b^4 + A*b^5)*d^5*e^3 + 4080*(2*B*a^2*b^3 + A*a*b^4)*d^4*
e^4 - 13260*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^5 + 12155*(B*a^4*b + 2*A*a^3*b^2)*d^2*
e^6 + 58344*(B*a^5 + 5*A*a^4*b)*d*e^7)*x^2 - (1536*B*b^5*d^7*e - 306306*A*a^5*d*
e^7 - 2176*(5*B*a*b^4 + A*b^5)*d^6*e^2 + 16320*(2*B*a^2*b^3 + A*a*b^4)*d^5*e^3 -
53040*(B*a^3*b^2 + A*a^2*b^3)*d^4*e^4 + 48620*(B*a^4*b + 2*A*a^3*b^2)*d^3*e^5 -
21879*(B*a^5 + 5*A*a^4*b)*d^2*e^6)*x)*sqrt(e*x + d)/e^7

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.353426, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)*(e*x + d)^(3/2),x, algorithm="giac")

[Out]

Done